\[r\]
with at any time \[t\]
the ant clockwise angle of the particle from the positive \[x\]
axis being given by \[\theta = \frac{t}{t+1}\]
/>br />
the angular velocity is \[\omega = \frac{d \theta }{dt} =\frac{(t+1) \frac{d(t)}{dt}- t \frac{d(t+1)}{dt}}{(t+1)^2}= \frac{1}{(t+1)^2}\]
using the quotient rule.
Similarly the angular acceleration is
\[\alpha = - \frac{2}{(t+1)^2}\]
.The distance moved is a time
\[t\]
is \[s= r \theta = =\frac{rt}{t+1}\]
, the speed is \[v= r \omega = \frac{r}{(t+1)^2}\]
and the acceleration is \[a= r \alpha = - \frac{2r}{(r+1)^3}\]
.As
\[t \rightarrow \infty, \; \theta \rightarrow 1\]
and \[\theta , \alpha \rightarrow 0\]
.